L(s) = 1 | − 2.88·3-s − 1.71·5-s − 4.47·7-s + 5.35·9-s − 4.06·11-s − 4.73·13-s + 4.96·15-s − 17-s − 8.07·19-s + 12.9·21-s + 4.47·23-s − 2.05·25-s − 6.79·27-s − 1.34·29-s + 5.63·31-s + 11.7·33-s + 7.68·35-s + 9.06·37-s + 13.6·39-s + 8.49·41-s + 0.753·43-s − 9.19·45-s − 4.24·47-s + 13.0·49-s + 2.88·51-s − 1.55·53-s + 6.98·55-s + ⋯ |
L(s) = 1 | − 1.66·3-s − 0.767·5-s − 1.69·7-s + 1.78·9-s − 1.22·11-s − 1.31·13-s + 1.28·15-s − 0.242·17-s − 1.85·19-s + 2.82·21-s + 0.934·23-s − 0.410·25-s − 1.30·27-s − 0.250·29-s + 1.01·31-s + 2.04·33-s + 1.29·35-s + 1.49·37-s + 2.19·39-s + 1.32·41-s + 0.114·43-s − 1.37·45-s − 0.619·47-s + 1.86·49-s + 0.404·51-s − 0.213·53-s + 0.941·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 - 5.63T + 31T^{2} \) |
| 37 | \( 1 - 9.06T + 37T^{2} \) |
| 41 | \( 1 - 8.49T + 41T^{2} \) |
| 43 | \( 1 - 0.753T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 61 | \( 1 + 8.96T + 61T^{2} \) |
| 67 | \( 1 - 2.40T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 6.99T + 73T^{2} \) |
| 79 | \( 1 + 0.499T + 79T^{2} \) |
| 83 | \( 1 + 0.659T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.37799215141721542571037379202, −6.56728455679561933243288013064, −6.22033867618674314845209632335, −5.49860430646856564235487885963, −4.60176613805979276168621908334, −4.27332520067681738231289442801, −3.06347340453918951905178581853, −2.35630154641327929987662769472, −0.57394683759065298038374972875, 0,
0.57394683759065298038374972875, 2.35630154641327929987662769472, 3.06347340453918951905178581853, 4.27332520067681738231289442801, 4.60176613805979276168621908334, 5.49860430646856564235487885963, 6.22033867618674314845209632335, 6.56728455679561933243288013064, 7.37799215141721542571037379202